Conjectured lower bound for near-maximal clique packings

Let $G(n,p)$ be the Erdős–Rényi random graph, let $p\in(0,1)$ and $C\in\mathbb{N}_{\geq 3}$ be constants, and let $k_0$ be the least integer such that the expected number of $k_0$-cliques in $G(n,p)$ is less than $1$. Set $k=k_0-C$. An edge-disjoint $k$-clique packing is a collection of $k$-cliques no two of which share an edge. Write $\gamma$ for the exponent governing the expected number of $k$-cliques, as in the source. Near-maximal clique-packing conjecture. With high probability, $G(n,p)$ contains at least

$$\frac{2(\gamma-2-o(1))pn^2\log n}{k^4}$$

edge-disjoint $k$-cliques. This conjecture concerns the expected duration of the random clique-removal process; the paper's method is intended to establish the corresponding lower bound, while the optimal packing size remains uncertain.